The class of 2-multigraphs with a prescribed degree sequence

We study both existence and structural properties of 2-multigraphs in the class of all 2-multigraphs with a prescribed degree sequence D. A 2-multigraph in defined to be parsimonious provided no 2-multigraph with the same degree sequence has fewer edges in its supporting (i.e., underlying) graph. In a parsimonious 2-multigraph G each connected component of the graph Gr formed by the edges of multiplicity 1 is either a star or a triangle. We attempt to understand the relationship between degree sequences and these graphs Gr .